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Richard Muller
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Prof Physics, UC Berkeley, author "Now, The Physics of Time"Upvoted by
José Ilhano Silva
, M. S. Mathematics & Differential Geometry, Federal University of Ceará (2017) and
Kathleen Nelson
, M.Ed. Mathematics & Education, University of Minnesota - Twin Cities (1988)5y
Originally Answered: What was your mathematical wall?

Up until graduate school, I found math to be relatively easy, with an exception when I accidentally took a course dominated by future Math professors. But that wasn’t a wall; it was simply difficulty in keeping up the speed with which they mastered the material. But then in graduate school, in my second year, I started learning the wonderful (indeed, even then I thought it was beautiful) math of complex variables. If you think of mathematical functions as extended into the complex domain, things simplify enormously. Using simple curves, you can calculate integrals that previously seemed impossible.

But there was something bothersome about this math. I couldn’t visualize it. I had no trouble with vector calculus; Gauss’ Law and Stokes’ Theorem, and all that, but now I was confronting analytic functions. These are complex functions which have the same derivative no matter which direction you approach the point in complex space. I could accept the definition, and I could even do OK on the problem sets and the exams, but I didn’t feel that I really understood them. They seemed magical. That made them powerful. But it was a magic that seemed to act beyond my comprehension.

I took a course from the great theoretical physicist Geoff Chew (he was also a former semi-professional baseball player who turned down an opportunity to join the major leagues because he wanted to do theoretical physics), in which he put forth his contention that the analyticity of the quantum mechanical object known as the S-matrix put such a fantastic constraint on the theory that it might be all that was needed to define quantum theory. Fascinating—but beyond my deeper understanding. The key, he said, was that the S-matrix was unitary. I knew that the S-matrix was unitary, but my understanding of that concept was not sufficiently deep for me to be able to see how powerful that idea was.

Analytic continuation still fascinates me, but that doesn’t mean that I really understand it very deeply. I know how to do it, and what some of its consequences are, but it is not a tool, something I use to think about reality. Most of our favorite functions can readily be continued into complex space in obvious ways. I can follow the work when others do something original with the concept, but I can’t imagine how they came up with the idea in the first place. I am at my wall.

Here’s my favorite example of analytic continuation. If we sum the squares of a series, such as f(x)=∑xn
, we can easily show that f(x)=11–x. Both equations give the same value for x < 1, so they seem to be describing the same function. But if you plug in x = 2, then the sum equation becomes f(2)=1+2+4+8+16+…=∞, but the analytic equation becomes f(2)=11–2=–1

. Thus the analytic equation “makes sense” of a function that actually doesn’t converge. Similar arguments are often made to confuse beginners into thinking that 1+2+3+4+… = -1/12.

Analytic continuation finds its greatest use when the equations extend into the complex plane. It is used extensively in much of modern physics field theory, as well as in mathematics; the famous Riemann hypothesis of number theory is actually a statement of the location of the zeros in an analytic continuation of a relatively simple function known as the Riemann zeta function.

I can follow this math. I know a lot about it. But I don’t feel that I truly understand it. I don’t see why it works. To me, vector calculus works because it makes so much sense; differential equations describe reality; Riemannian geometry with curved 4D space time is something I am comfortable with; even topology results, often surprising (I wouldn’t have imagined that one could turn a spherical shell inside-out without making any creases!), can often be visualized. But complex functions and their mysterious behavior still mystify me.
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